how to deal with negative powers as mentioned in one of the practice question attached with the question.
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Negative Exponents
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Work from the INSIDE OUT:If x and y are nonzero integers, is (xˉ¹ + yˉ¹)ˉ¹ > [(xˉ¹)(yˉ¹)]ˉ¹ ?
(1) x = 2y
(2) x + y > 0
xˉ¹ + yˉ¹ = 1/x + 1/y = (x+y)/xy.
xˉ¹yˉ¹ = 1/(xy).
Substituting the resulting expressions back into the question stem, we get:
[ (x+y)/xy ]ˉ¹ > [ (1/xy) ]ˉ¹
xy/(x+y) > xy.
Question stem, rephrased:
Is xy/(x+y) > xy?
Statement 1: x=2y
Substituting x=2y into xy/(x+y) > xy, we get:
(2y*y)/(2y+y) > (2y*y)?
2y²/3y > 2y² ?
Since y is nonzero, 2y²>0.
Thus, we can safely divide each side by 2y².
The question stem then becomes:
1/(3y) > 1?
Since y is a nonzero integer, 1/(3y) is equal to a positive or negative fraction between -1 and 1.
Thus, the answer to the question stem is NO:
It is not possible that 1/(3y) > 1.
SUFFICIENT.
Statement 2: x+y > 0
Case 1: x=y=1
Substituting x=y=1 into xy/(x+y) > xy, we get:
(1*1)/(1+1) > 1*1
1/2 > 1.
NO.
Case 2: x=-1 and y=3
Substituting x=-1 and y=3 into xy/(x+y) > xy, we get:
(-1*3)/(-1+3) > -1*3
-3/2 > -3.
YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.
The correct answer is A.
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Hi duam12,
Mitch's explanation focused on the algebra behind the question. Here's a way to deal with this question that focuses on the Number Properties...
Since Mitch already showed the steps behind translating the question, we have this...
Is XY/(X+Y) > XY? This is a YES/NO question. We are told that X and Y are non-0 integers
Fact 1: X = 2Y
This tells us that X and Y are either both positive OR both negative.
That means that XY = a positive integer. Plugging this into the question, we have
Is (pos int)/(X+Y) > (same pos. int)?
If X and Y are both positive, then the answer is NO (since the number on the "left" will decrease due to dividing by (X+Y)).
If X and Y are both negative, then the answer is NO (since the number on the "left" will become negative).
Fact 1 is SUFFICIENT
Fact 2: X + Y > 0
Here we know that we'll have either 2 positives values or 1 positive and 1 negative (and the positive will more than "offset" the negative).
If X and Y are both positive, then we know from Fact 1 that the answer will be NO
If we have 1 positive and 1 negative, then....
XY = negative integer
X+Y = positive
and the question becomes...
Is (neg int)/(pos >=1) > (same neg int)?
If X + Y is ANYTHING greater than 1, then the "left" side will be greater than the "right" side and the answer is YES.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
Mitch's explanation focused on the algebra behind the question. Here's a way to deal with this question that focuses on the Number Properties...
Since Mitch already showed the steps behind translating the question, we have this...
Is XY/(X+Y) > XY? This is a YES/NO question. We are told that X and Y are non-0 integers
Fact 1: X = 2Y
This tells us that X and Y are either both positive OR both negative.
That means that XY = a positive integer. Plugging this into the question, we have
Is (pos int)/(X+Y) > (same pos. int)?
If X and Y are both positive, then the answer is NO (since the number on the "left" will decrease due to dividing by (X+Y)).
If X and Y are both negative, then the answer is NO (since the number on the "left" will become negative).
Fact 1 is SUFFICIENT
Fact 2: X + Y > 0
Here we know that we'll have either 2 positives values or 1 positive and 1 negative (and the positive will more than "offset" the negative).
If X and Y are both positive, then we know from Fact 1 that the answer will be NO
If we have 1 positive and 1 negative, then....
XY = negative integer
X+Y = positive
and the question becomes...
Is (neg int)/(pos >=1) > (same neg int)?
If X + Y is ANYTHING greater than 1, then the "left" side will be greater than the "right" side and the answer is YES.
Fact 2 is INSUFFICIENT
Final Answer: A
GMAT assassins aren't born, they're made,
Rich
